Solve frac{sqrt{14}+2}{1-sqrt{71}}

$$\frac { \sqrt { 14 } + 2 } { 1 - \sqrt { 71 } }$$

Evaluate

$$-\frac{\sqrt{14}}{70}-\frac{\sqrt{71}}{35}-\frac{\sqrt{994}}{70}-\frac{1}{35}\approx -0.773167464$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{14}+2}{1-\sqrt{71}} by multiplying numerator and denominator by 1+\sqrt{71}.

$$\frac{\left(\sqrt{14}+2\right)\left(1+\sqrt{71}\right)}{\left(1-\sqrt{71}\right)\left(1+\sqrt{71}\right)}$$

Consider \left(1-\sqrt{71}\right)\left(1+\sqrt{71}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(\sqrt{14}+2\right)\left(1+\sqrt{71}\right)}{1^{2}-\left(\sqrt{71}\right)^{2}}$$

Square 1. Square \sqrt{71}.

$$\frac{\left(\sqrt{14}+2\right)\left(1+\sqrt{71}\right)}{1-71}$$

Subtract 71 from 1 to get -70.

$$\frac{\left(\sqrt{14}+2\right)\left(1+\sqrt{71}\right)}{-70}$$

Apply the distributive property by multiplying each term of \sqrt{14}+2 by each term of 1+\sqrt{71}.

$$\frac{\sqrt{14}+\sqrt{14}\sqrt{71}+2+2\sqrt{71}}{-70}$$

To multiply \sqrt{14} and \sqrt{71}, multiply the numbers under the square root.

$$\frac{\sqrt{14}+\sqrt{994}+2+2\sqrt{71}}{-70}$$

Multiply both numerator and denominator by -1.

$$\frac{-\sqrt{14}-\sqrt{994}-2-2\sqrt{71}}{70}$$